On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex
The authors study a discrete-time map \(T:\mathbb{R}^n\to\mathbb{R}^n\) where the \(i\)-th component of \(T\) is \[ T_i(x)=\frac{(1+r_i)(1-c_i)x_i}{1+\sum_{j=1}^nb_{ij}x_j}+c_ix_i, \] and \(0< c_i <1\); \(b_{ij},r_i>0\). They refer to this as the generalized competitive Atkinson-Allen map, which reduces to the classical map when \(r_i=1\) and \(c_i=c\) for all \(i\). The authors prove that all such maps have a simplex \(\Sigma\) which is a globally attracting invariant manifold of dimension \(n-1\). Concentrating on the \(n=3\) case they show that the dynamics on \(\Sigma\) can be grouped into 33 distinct cases, and partially analyse the dynamics in all of these cases. They find that the dynamics of the generalized competitive Atkinson-Allen model is more complex than that of the standard model.
scientific article; zbMATH DE number 6806961
scientific article